Resonant wave interaction with random forcing and
dissipation
Paul A. Milewski
Department of Mathematics
University of Wisconsin, Madison
Esteban G. Tabak and Eric Vanden Eijnden
Courant Institute of Mathematical Sciences
New York University
December 14, 2000
Abstract
A new model for studying energy transfer is introduced. It consists of a \resonant duo" {a
resonant quartet where extra symmetries support a reduced subsystem with only two degrees
of freedom{, where one mode is forced by white noise and the other one is damped. This
system has a single free parameter: the quotient of the damping coecient to the amplitude
of the forcing times the square root of the strength of the nonlinearity. As this parameter
varies, a transition takes place from a Gaussian, high-temperature, near equilibrium regime,
to one highly intermittent and non Gaussian. Both regimes can be understood in terms of
appropriate Fokker-Planck equations.
Keywords: Resonance, intermittency, wave turbulence, invariant measures
1
1 Introduction
Many systems in Nature receive and dissipate energy in very dierent scales, behaving like conservative systems in between. Hence energy is permanently transferred through this intermediate,
inertial range, which often includes many decades of spatial and temporal scales. Typically, the
path among scales that this ux of energy adopts is not deterministic and orderly, but rather chaotic
and noisy. When this is the case, the ux is said to occur through a turbulent cascade of energy. The
most famous example of such a cascade lies in the eld of isotropic uid turbulence, widely studied
since the pioneering work of Kolmogorov. Many others, however, take place in Nature. The ocean,
for instance, displays a very rich set of highly anisotropic cascades, whose variation with scale is
thought to be determined by a changing balance between the eects of rotation, stratication, and
wave breaking.
For dispersive systems, such as surface and internal waves in the ocean, energy transfer among
scales is thought to occur largely through resonant sets, typically triads or quartets. A rich body of
theory has been developed concerning such systems, under the name of Wave (or Weak) Turbulence.
The theory predicts kinetic equations for the evolution of the energy spectrum, and self-similar stationary solutions to these equations ([1], [2], [3], [4].) However, the complexity of the systems under
study make many of the hypothesis underlying these results necessarily heuristic. In fact, recent
studies of a simple, one-dimensional model for dispersive waves, show an incredibly rich behavior,
with a number of [often coexistent] self-similar spectra, some of them apparently inconsistent with
the existing theory ([5], [6]).
Our purpose here is to consider even simpler models of energy transfer, involving as few modes as
possible, in order to isolate the roots of this variety of regimes. To this end, starting from a relatively
general dispersive system, we single out a resonant quartet, with two modes forced by white noise
and two damped. Then we invoke a symmetry of the quartet equations to reduce the system even
further, to a system that we call a \forced and damped resonant duo", which exhibits, when freed
from forces and dissipation, the Hamiltonian structure and conserved quantities that characterize
far more complex dispersive systems. This system is so reduced though, that its numerical solution
is quite straightforward, and much can be understood even on purely theoretical grounds. Yet
the system's behavior is surprisingly rich, with a transition between Gaussian, near-equilibrium
2
behavior, to highly intermittent. This suggests strong analogies to similar, largely unexplained
transitions in much more complex systems.
The plan of this paper is the following. After this introduction, in section 2, we describe the main
features of a resonant quartet, and justify the introduction of white noise as the most controllable
energy source. In section 3, we reduce the system even further, to a forced and damped resonant
duo, and derive some expected values and bounds for its statistically steady states. In section 4, we
solve the reduced system numerically, and show the existence of two distinct regimes: one Gaussian
and close to equilibrium, the other highly intermittent and non Gaussian. In sections 5 and 6,
the [approximate] invariant measures for both regimes are explicitly obtained as [near] solution to
the system's Fokker-Planck equation. The mechanisms underlying transport of energy in the two
regimes are discussed in section 7. Finally, in section 8, we summarize our conclusions, and suggest
some further work.
2 A Forced and Damped Resonant Quartet
For concreteness, we shall start with a one-dimensional partial integro-dierential equation of the
form
i @@t = L + jj plus forcing and dissipation;
(2.1)
where L is an Hermitian linear operator with symbol L^ = !(k). In the inertial range, this system
2
can be written in the Hamiltonian form
i @@t = H
where
H=
Z
Z
^
!(k)j(k)j dk + 2 j(x)j dx:
2
4
Even this relatively simple one-dimensional system, with ! = jkj = and a slightly more general
nonlinearity, has been recently shown to display a very rich and puzzling phenomenology, with a
number of self-similar statistically steady states [6]. These states often coexist, occupying disjoint
ranges in Fourier space, while sometimes one of them takes over the whole inertial range. The
underlying bifurcations appear to depend very delicately on the nature and strength of the forcing
and dissipation, on the sign of the parameter tuning the nonlinearity, and, since all numerical
1 2
3
experiments take place in nite domains, on the size of these domains (or, correspondingly, on the
spacing between modes in Fourier space.)
Our goal here is to isolate a simpler subsystem of (2.1) where the issue of energy transfer among
modes is more transparent. To this end, we shall consider a single resonant quartet, i.e. a set of
four modes ^ j , such that the resonant conditions
k +k = k +k
! +! = ! +!
1
4
2
1
4
2
3
3
are satised. When this is the case, and those four modes are the only ones excited {at least to
leading order{ in the initial conditions, the ^ j 's are approximated, in the limit of small amplitudes,
P
by ^ j (t) = aj ( )e?i ! ? m t , where = t, m = j jaj j and the a's obey the resonant
equations ([7], [8])
i da
d = 2 a a a ? ja j a
i da
(2.2)
d = 2 a a a ? ja j a
i da
d = 2 a a a ? ja j a
i da
d = 2 a a a ? ja j a
( j
2
)
4
=1
2
1
2
3
4
4
2
3
1
3
4
1
2
2
4
1
3
1
2
3
4
2
2
2
2
2
1
2
3
4
These equations are also Hamiltonian, with
?
H = 4 cos() ? 2 + + + :
1
2
3
4
1
4
4
2
4
3
(2.3)
4
4
Here aj = j e and = + ? ? . In addition to preserving H , the solutions satisfy the
\Manley-Rowe" relations
dja j = dja j = ? dja j = ? dja j ;
(2.4)
d
d
d
d
P
P
from which conservation of mass m = j jaj j , momentum p = j kj jaj j and linear energy
P
e = j !j jaj j , follow. In fact, one can solve these equations analytically. Setting
j
1
4
2
1
3
2
4
2
4
=1
4
=1
2
2
3
4
=1
2
2
ja j ( )
ja j ( )
ja j ( )
ja j ( )
1
2
3
1
2
2
2
2
= X ( ) + c ;
= X ( ) + c ;
1
2
= ?X ( ) + c ;
= ?X ( ) + c ;
3
4
4
2
2
where c + c ? c ? c = 0, one obtains, after some manipulation,
1
2
3
4
X ( ) = cos(
+ ) + ;
with , , and determined by the initial data.
In order to study energy transfer, one needs to force some of the modes of the system (2.2), and
dissipate some others. In order to control the amount of energy going through the system, it is best
to force it through white noise, since, when a system is forced deterministically, it is not clear a
priori how much energy the forces provide. Typically, such systems will reach equilibrium even in
the absence of dissipation. A single nonlinear oscillator forced sinusoidally, for instance, will reach
a steady state by detuning from the frequency of the forcing. Hence thinking of deterministic forces
as permanent energy sources is not necessarily accurate. On the other hand, when either the forces
or the system become more irregular (the latter case arising when the system's internal dynamics is
chaotic), the forces do behave systematically as an energy source. In the extreme example provided
by white noise, the amount of energy input to the system is strictly controllable, as the following
theorem shows:
Theorem : Consider a dynamical system of the form
dui = F (u; t) + w_ ;
i i
dt
P
where w_ i stands for white-noise. Then the energy of the system, E = i juij2, evolves in the
following, separable way:
d hE i = hE (u; t)i + E ;
d
w
dt
where Ed(u; t) is the deterministic rate of energy change, that would take place even without the
P
white noise, and Ew = i2 (the brackets in the expressions above represent ensemble averages).
Thus, for instance, if the unforced system is conservative, Ed is zero, and the energy grows linearly
in time. If, on the other hand, the unforced system includes dissipation, and if a state of statistical
equilibrium is achieved, then we must have hEd(u; t)i + Ew = 0.
Proof : This theorem is a simple corollary of Ito Calculus. For readers unfamiliar with it, we
prove a discrete analogue for the system
p
uni ? uni = Fi(un; n) t + i win t ;
+1
5
(2.5)
where hwini = 0, winwjm = ij nm , and the total energy is dened to be
En =
From (2.5) and (2.6), we obtain
n+1
E
?E
n
*
= Re
= Re
=
X?
i
*
uni ?
X
X
i
+1
X
i
junij :
(2.6)
2
+
?
un un+1 + un
i
i
i
np
p
n
Fi (un; n) t + i wi t 2 uni + Fi(un; n) t + i wi t
i D
Fi (un; n)(2 uni + tFi (un; n))
Re
E
+
+ i t
2
= (hEd (u; t)i + Ew )t ;
which concludes the proof. For instance, if Fi(un; n) consists of an energy preserving part plus
dissipative terms of the form ?i uni, then
hE n ? E ni =
+1
X?
i
i ? 2ijuij t +
2
2
X
i
i juij (t)
2
2
2
and, in the continuous limit,
dhE i = X ? ? 2 ju j :
i i
i
dt
i
Thus, for a single, nonlinear damped oscillator forced by white noise,
2
2
i ddt = (! + F (j j) ? i ) + w_ ;
(2.7)
(F real), if a statistically steady state is reached, it necessarily satises
j j = 2 :
2
2
Based on these considerations, we shall study the following generalization of (2.2):
i ddat
i ddat
i ddat
i ddat
1
= 2 a a a ? ja j a + w_ (t);
2
= 2 a a a ? ja j a ? i a ;
3
= 2 a a a ? ja j a ? i a ;
4
= 2 a a a ? ja j a + w_ (t) ;
4
3
2
1
2
4
4
2
3
1
1
2
1
3
3
4
6
2
2
2
2
1
2
3
4
1
2
3
4
(2.8)
where w_ (t) and w_ (t) represent white noise. The reasoning behind the new terms added to (2.2) is
the following: Once one has decided to force one of the modes {say a for concreteness{ with white
noise, then a needs to be forced also {and with white noise of the same amplitude{ if there is to be
any hope for the system to reach statistical equilibrium (for otherwise the Manley{Rowe relations
(2.4), combined with the theorem above applied to the equations for a and a separately, imply
that hja j ? ja j i will necessarily diverge.) Similarly, once a is damped, so should a . Here it is
not crucial that the two damping coecients be equal, but there does not seem to be much point in
breaking the system's symmetry by deciding otherwise (in fact, we shall use this symmetry below
to reduce our model even further.)
1
4
1
4
1
1
2
4
2
4
2
3
3 Reduction to a Forced and Damped Duo
The symmetries between a and a and between a and a in the system (2.8) suggest a further
reduction to two variables, by using a single random process for w (t) and w (t), and considering
initial data such that a = a and a = a . After renaming the variables, one ends up with the
system for a resonant duo:
1
4
2
3
1
1
4
2
4
3
i ddat = 2 a a ? ja j a + w_ (t);
(3.1)
(3.2)
i ddat = 2 a a ? ja j a ? i a :
A disclaimer seems appropriate here: resonant quartet equations such as (2.8) arise naturally
as asymptotic reductions of larger systems, when only a handful of modes is initially excited. The
resonant duo proposed here, on the other hand, represents only a particularly symmetric instance
of a resonant quartet. It should not be considered as a reduced model for the interaction among
only two modes in a larger system, since the implication would be that the two modes have the
same wavenumber and frequency. This is not totally unthinkable, since many systems are indexed
by wavenumber (with corresponding frequency) and something else, but it is certainly not the case
of models such as (2.1), where each wavenumber points to a single degree of freedom.
When both and are set to zero, the system in (3.1, 3.2) is Hamiltonian, with
1
2
2
1
1
2
2
1
2
2
2
2
1
2
2
?
?
H = a a + a a ? 2 ja j + ja j ;
2
2
2
1
2
1
2
2
1
7
4
2
4
and it has exact solutions of the form
ja j = cos(
t + ) + ja j = ? cos(
t + ) + 1
2
2
1
2
2
Notice that there is a single non-dimensional parameter in (3.1, 3.2):
D = p :
(3.3)
Thus all but one of , , or can be made equal to one by a suitable rescaling of time and
amplitudes, and D serves as a single control parameter for the system.
In the remaining of this paper, we shall study the properties of the statistically steady solutions
to (3.1, 3.2). If such state is achievable, the theorem in the previous section, applied to the full
system, implies that
ja j = 2 ;
(3.4)
2
2
2
which states that the system's energy input needs to be fully dissipated by the damping of the
second oscillator. The same theorem applied to either of the two equations alone, on the other
hand, yields
?4 ja j ja j sin(2) = ;
(3.5)
2
1
2
2
2
2
where = ? , and we are writing aj = j ei . The left hand side of this equation represents
nonlinear energy transfer among the two modes, which has to equal the total energy input from
white noise. To obtain a lower bound for ja j, one derives, from equation (3.2), the identity
d log ?ja j = ?4 ja j sin(2) ? 2
dt
which, after taking averages and looking for a statistically steady state, yields
ja j 2 :
(3.6)
Another conjectured lower bound for a , of a more \thermodynamical" nature, states that
ja j ja j = 2 ;
(3.7)
since a situation in which forcing and dissipation on an otherwise symmetric duo should yield a
higher mean square amplitude for the dissipated mode than for the forced one would contradict the
second principle of thermodynamics (energy would be transferred \up" from the less to the more
excited state).
2
j
1
1
2
2
1
1
2
2
1
1
2
2
8
2
2
0.16
0.14
0.12
2
<|a1,2| >
0.1
0.08
0.06
0.04
0.02
0
0
0.5
1
1.5
2
2.5
D
3
3.5
4
4.5
5
Figure 1: hja j i (stars) and hja j i (circles) as functions of D = =p . Also plotted are the
p
two lower bounds (3.6) and (3.7) in solid line, and hja j i = = 3 (dotted). The results are
averages over time and about 800 realizations of white noise from xed initial data a = 0:1 + 0:2i,
a = 0:15 ? 0:1i, with time averaging from t = 2500 to t = 5000.
1
2
2
2
1
2
1
2
4 Numerical Simulations
Simulating numerically the system in (3.1, 3.2) is a rather straightforward task. Since our interest
here lies in energy transfer over long time intervals, a simplectic procedure appears appropriate.
We have adopted a simplectic second order (implicit) Runge{Kutta for the deterministic part of
the system, and alternated it with an explicit addition of white noise. Thus the algorithm becomes
?
2
a = a n ? i t 2 ah
? a = a n ? i t 2 ah
p
an = a + t wn
an = a ;
1
1
2
2
2
1
2
?
+1
+1
2
1
ah ? jahj ah
ah ? jahj ah ? i ah
1
1
2
2
2
2
1
2
2
(4.1)
1
2
where ahj = anj + aj and wn is a random complex variable drawn from a Gaussian distribution
with variance one.
Figure 1 shows the results of a series of experiments, in which has been kept xed at = 1,
the amplitude of the white noise at = 0:02, and has been varied from = 0:00125 to = 0:1,
1
2
9
0.14
0.12
0.1
|a1,2|
2
0.08
0.06
0.04
0.02
0
2020
2040
2060
2080
2100
t
2120
2140
2160
2180
Figure 2: An individual realization of the resonant duo, with D = 0:25. The two modes a and
a oscillate around each other much as in the unforced, undissipated system, but with less orderly
paths.
1
2
which corresponds to the non-dimensional parameter D in (3.3) ranging from D = 0:0625 to D = 5.
We have typically run 800 realizations of white noise from xed initial data (a = 0:1 + 0:2i,
a = 0:15 ? 0:1i) up to t = 5000, and computed time (as well as ensemble) averages from t = 2500
(longer times and more realizations where needed for the very large and very small values of , for
reasons that will become clear below), with t = 0:0025. We have plotted hja j i with stars, hja j i
with circles, and the two lower bounds for hja j i from (3.6) and (3.7) in solid line (the second lower
bound agrees, of course, with the expected value of ja j from (3.4).) In addition, there is a dotted
p
line, corresponding to hja j i = = 3 , that will be explained below.
We can see the nearly perfect agreement of the numerical results for hja j i with their exact value
(3.4), and the sharp nature of the two bounds (3.6) and (3.7), which all but dene the dependence
of hja j i on the parameters , , and . There are clearly two dierent regimes, depending on
which lower bound is enforced. For small values of D, hja j i hja j i, and we are close to
thermodynamical equilibrium. For large values of D, on the other hand, hja j i hja j i, and the
sharp nature of the lower bound (3.7) suggests that the relative phase of the two oscillators is locked
near = ?=4 whenever a is active. In fact, the dotted line, which ts very well the asymptotic
1
2
1
1
2
2
2
1
2
2
2
1
2
2
2
1
2
2
2
1
2
10
2
2
2
2
0.4
0.35
0.3
2
<|a1,2| >
0.25
0.2
0.15
0.1
0.05
0
3400
3600
3800
4000
4200
4400
4600
4800
t
0.35
0.3
0.25
2
<|a1,2| >
0.2
0.15
0.1
0.05
0
4100
4120
4140
4160
4180
4200
t
4220
4240
4260
4280
4300
Figure 3: An individual realization of the resonant duo, with D = 5. ja j is essentially zero most
of the time, with intermittent, brief outbursts of energy. a) A relatively long interval, with a few
transfer events. b) Detail of one the events.
2
11
2
10
1
p(|a1|2)
10
0
10
−1
10
0
0.02
0.04
0.06
0.08
0
0.02
0.04
0.06
0.08
|a1|2
0.1
0.12
0.14
0.16
0.18
0.1
0.12
0.14
0.16
0.18
2
10
1
p(|a2|2)
10
0
10
−1
10
|a2|2
Figure 4: Numerical invariant measure for D = 0:25, and Gaussian measures with the same variance. The distributions of both a and a are indistinguishable from [complex] Gaussian, and their
variances are equal.
1
2
behavior of hja j i, corresponds to a value = ?=6, which will be shown below to arise from
simple theoretical considerations.
Figures 2 and 3 display individual realizations of the two regimes, corresponding to D = 0:25
and D = 5. In the former, the two modes oscillate around each other much as in the unforced,
undissipated system, but with more noisy paths. In the latter, ja j is essentially zero most of the
time, with intermittent, brief outbursts of energy.
Figures 4 and 5 show the (numerical) invariant measures for the same two values of D, contrasted
with Gaussian measures with the same variance. We see that, for D = 0:25, the distributions of
both a and a are indistinguishable from Gaussian, while for D = 5, a is still Gaussian, but a is
fundamentally dierent, with a sharp peak at ja j = 0 and a very long tail.
These behaviors will be fully accounted for in sections 5, 6, and 7.
1
2
2
1
2
1
2
12
2
2
10
1
1
2
p(|a | )
10
0
10
−1
10
0
0.02
0.04
0.06
0.08
|a |2
0.1
0.12
0.14
0.16
0.18
0.05
0.06
0.07
0.08
0.09
1
10
10
0
2
p(|a2| )
10
−10
10
−20
10
0
0.01
0.02
0.03
0.04
2
|a2|
Figure 5: Numerical invariant measure for D = 5, contrasted with Gaussian measures with the
same variance. a is still Gaussian, but a is fundamentally dierent, with a sharp peak at ja j = 0,
and a very long tail.
1
2
2
5 The Near-Equilibrium, High-Temperature Regime
The system (3.1, 3.2) has four real degrees of freedom, which can be taken to be the amplitudes and
phases of the two modes. However, only the dierence between the two phases, = ? enters
the dynamics of the amplitudes, so the system can be further reduced to the three-dimensional one
2
d = 2 sin(2) + + p w_
dt
4
2
d = ?2 sin(2) ? dt
d() = ? ? (1 + 2 cos(2)) + p w_ ;
dt
2
1
2
2
2
2
1
2
1 2
2
2
1
1
2
2
1
2
1
1
(5.1)
(5.2)
(5.3)
where w_ and w_ stand for two independent real white-noises.
The corresponding Fokker-Planck operator is
1
2
L = LH + LF
13
(5.4)
where
LH
= 2 sin(2) @@ ? @@
?
+ ? (1 + 2 cos(2)) @
@ 1
2
2
2
2
1
1
2
(5.5)
2
1
represents the Hamiltonian component of the evolution, and
@ + @
LF = 4 @@ ? @@ + 4 @
(5.6)
4 @ ()
represents the eects of forcing and damping. An invariant measure of the evolution f ( ; ; )
solves
L f = 0 ;
(5.7)
2
2
1
2
1
2
2
2
1
2
2
1
2
2
1
2
where the operator L is the adjoint of L.
Let us consider rst the regime D 1, for which the numerical experiments suggest a near
Gaussian invariant measure with h i h i = =2 , that is,
2
1
2
2
2
f ( ; ; ) = C e? (21 22 )=2 :
1
2
1
2
2
+
(5.8)
We now show that the density in (5.8) is indeed the leading order term of an expansion in D. Upon
rescaling
! p ~ ; ! p ~ ; t ! t~=;
and dropping the tildes, the equation in (5.7) reduces to
1
1
2
2
1 L + L f = 0 ;
D H F
2
(5.9)
where the operators L H and L F are LH and LF with , , set to one. We look for a solution of
(5.9) of the form
f = f + D f + O(D ):
(5.10)
2
0
4
1
Inserting this expansion into (5.9) and equating coecients of equal powers of D, we obtain
L H f = 0;
L H f = ?L F f :
0
1
14
0
(5.11)
(5.12)
(5.11) implies that f belongs to the null-space of L H , f 2 Ker L H . Since the Hamiltonian dynamics
conserves both the Hamiltonian and the energy
?
H = ? 12 + + 2 cos(2); E = + ;
the null-space of L H is spanned by functions of the type times an arbitray function of E and
H , i.e. (5.11) yields
f = g (E; H ) ;
(5.13)
0
0
4
1
4
2
2
1
2
2
2
1
1
0
1
2
2
2
2
where g() is arbitrary except for the boundary conditions for (5.11),
lim f = 0;
21 +22 !1
lim f = 0;
1 !0
0
lim f = 0:
1 0
We determine g() from the solvability condition for (5.12),
2 !0
2 0
L F f 2 Ran L H = (Ker L H )?:
(5.14)
0
By an argument similar to the one above, it follows that the null-space of L H is spanned by the
functions of the form h(E; H ), where h() is arbitrary. Thus, the solvability condition in (5.14) is
given by
Z Z
0=
h(E; H )L F ( g (E; H )) d d d():
2
2
1
R+
0
2
1
2
We claim that this integral equation can be transformed into a partial dierential equation in E , H
for g(E; H ). To see this, change the integration variables in order to integrate rst on the regions
where E and H are constant, then on E , H . Since both h(E; H ) and g(E; H ) depend only on E
and H , the rst integral can be performed explicitly, and the resulting integral equation can be
written as
Z 1 Z E2=
0=
J (E; H )h(E; H )L g(E; H )dH dE:
(5.15)
4
1
?3E 2 =4
0
where J is some Jacobian and L is an operator in E , H whose explicit form must be obtained by
integrating L F ( ) at E , H constant. Since h(E; H ) is arbitrary in (5.15), the factor L g(E; H )
must be identically zero, which gives indeed a partial dierential equation in E , H for g(E; H ). The
full operator L is rather complicated and we shall not write it here explicitly, since we only need
its restriction on functions depending only on E . Indeed, under the consistent assumption that g
depends on E alone, the integral in (5.15) reduces to
1
1
2
1
1
0=
Z 1
0
h (E ) 4Eg(E ) + 2(E + E )g0(E ) + E g00(E ) dE;
2
15
2
(5.16)
where
h (E ) =
Z E 2 =2
?E 2 =2
h(E; H )J (E; H )dH;
Since h is arbitrary, (5.16) is equivalent to the dierential equation
4Eg(E ) + 2(E + E )g0(E ) + E g00(E ):
2
2
whose only bounded solution is g(E ) = Ce? E . It follows that
2
f = C e? (21 22 );
0
1
2
2
(5.17)
+
which in the original dimensional variables yields (5.8).
6 The Intermittent Regime: An Exactly Solvable Model
From the numerical experiments, we know that, when D 1, we reach a highly intermittent
regime, with on the average and a nearly locked phase . Notice that this is consistent
with equation (5.3), when it is dominated by the deterministic part. The locked phase then needs
to satisfy
1 + 2 cos(2) = 0 ;
1
2
so
p
sin(2) = 23 :
Moreover, only the phase yielding the sine with the minus sign is stable under the [deterministic]
dynamics of (5.3). This suggests replacing the system (5.1, 5.2, 5.3) by the reduced
d = ?2 + + p w_
dt
4
2
d = 2 ? :
dt
1
2
2
2
2
1
1
2
1 2
1
2
(6.1)
(6.2)
p
Here the constant , with 0 3=2, measures the eectiveness of phase locking, and should
p
approach the value 3=2 as D goes to innity. Equivalently, this amounts to saying that, in the
limit of large D, the invariant measure for the original system (5.1, 5.2, 5.3) satises
f ( ; ; ) f( ; )( + =6);
1
2
1
16
2
(6.3)
where () is the delta function and f( ; ) is the invariant measure for the approximate system
in (6.1, 6.2).
For this model, it is still true that, if a statistically steady state is achieved, it must have
1
2
= 2 :
2
2
(6.4)
2
Moreover, from
d log ? = ?4 ? 2 ;
dt
one obtains, instead of a lower bound for as in (3.6), the sharper result that
2
2
2
1
1
:
= 2
Finally, looking at the energy transfer among modes, one obtains
2
(6.5)
1
?4 = ;
2
1
2
2
(6.6)
2
which, together with (6.4) and (6.5), implies that
2 2
= 2 2
1 2
;
1
(6.7)
2
strongly suggesting that and are independent random variables.
In fact, one can nd an exact invariant measure for the system in (6.1, 6.2) satisfying these
properties. The Fokker-Planck operator for the new system in (6.1, 6.2) is given by
1
2
@ ;
L = 2 @@ ? @@ + 4 @@ ? @@ + 4 @
which, consistently with (6.7), has a separable invariant measure of the form
1
2
2
1
1
2
f( ; ) = C ?
1
2
1
2
1
2
1
2
2
(6.8)
2
1
2
2 = 2 e?2 (21 +22 )= :
(6.9)
1+2
2
p
The higher temperature mode, , is Gaussian, as observed in the numerics. For =2
the distribution for the lower temperature mode, , is essentially a power law with exponent
= ?1 + 2 = = ?1 + 2=D , which approaches ?1 as D goes to innity. This is consistent
with the observed sharp peak at = 0 and very long tail, since =2 h i = =2 for large D.
In fact, this distribution agrees nearly to perfection with the observed one, as the plots in Figure
1
2
2
2
2
2
2
2
2
17
2
4
10
3
10
2
2
p(|a2| )
10
1
10
0
10
−1
10
−2
10
−5
−4
10
−3
10
−2
10
|a |2
−1
10
10
2
Figure 6: Numerical invariant measure for ja j, D = 5, compared with the theoretical prediction in
(6.9). The value of 0:82 was drawn from the observed mean of , and then used to build the
distribution for .
2
1
2
6 show for D = 5. Here the value of 0:82 was drawn from the observed mean of , and then
used to build the distribution for .
The intermittent behavior of individual realizations is also easier to understand in the simplied
model in (6.1, 6.2). Notice that, whenever is smaller than =2 , equation (6.2) predicts damping
of . Hence the situation is the following: is pinned near zero most of the time by the strong
damping, while undergoes free Brownian motion. However, as soon as crosses the threshold
of instability = =2 , grows explosively, and soon starts to drag down, back to the stable
regime. White noise is not important during this bursts of energy transfer, so we can set = 0.
With this, the system in (6.1, 6.2) is exactly solvable; its solution for the initial condition (0) = r ,
(0) = r is given implicitly by
Z r1
1
dz
t = 2
;
1 z (r + r ? z + (= ) ln(z=r ))
ln( =r );
= r + r ? + (6.10)
1
2
2
1
2
2
1
1
2
1
2
1
1
2
1
2
2
2
2
2
2
2
2
1
2
1
2
1
2
1
1
1
A plot comparing this solution with the energy outburst of gure 3b is shown in Figure 7 (we picked
the initial values for (6.10) to t and at time t=4185). The very close agreement between
1
2
18
0.35
0.3
0.25
2
ρ1,2
0.2
0.15
0.1
0.05
0
4190
4195
4200
4205
t
4210
4215
4220
Figure 7: Comparison of the exact solution (6.10) with the energy outburst of gure 3b. The initial
values for (6.10) were picked to t and at time t=4185.
1
2
the two curves leaves little doubt that the scenario just described applies to the original equations
in (5.1, 5.2, 5.3) as well. Notice incidentally that the threshold of instability = =2 yields
the mean value h i in the intermittent regime; hence h i settles at a value such that = 0 is
neutrally stable.
2
1
2
1
2
1
2
7 Discussion
The behaviors observed in sections 5 and 6 can be explained by comparing various time-scales or,
equivalently, various rates associated with transport of energy in the system. The rst one is ,
giving the rate at which energy is removed from the system by means of dissipation. We shall
compare with
? = 4 j sin(2)j = + :
(7.1)
2
1
Since
2
2
2
1
2
2
d = 4 sin(2) + + p2 w_ ;
dt
d = ?4 sin(2) ? 2 ;
dt
2
1
2
2
2
1
2
2
2
1
2
2
2
1
2
2
19
1
? gives a measure of the [averaged] rate at which energy is transported between the modes by
means of the Hamiltonian part of the dynamics, normalized by the total energy of the system.
In the Gaussian regime, we have h i h i = =2 and
2
1
2
2
2
;
4 hj sin(2)ji 4 h ih ihj sin(2)ji 2
2
1
2
2
2
1
which, since D 1, implies
4
2
2
2
:
? 2
2
It follows that any blob of energy fed into the system on either of the modes will bounce back and
forth between them many times by means of oscillations before being dissipated. In other words,
the system is able to \thermalize" the modes, h i h i, and the actual temperature is xed by
the amount of forcing and damping applied to the system. In fact, consistently with this picture,
the Gaussian measure in (5.8) might also have been predicted by a rough application of equilibrium
statistical mechanics as the least biased measure given the information in the conserved quantity,
+ [9].
In the intermittent regime, one observes
2
1
2
1
2
2
2
2
p
h i = =2 h i = 3;
2
2
2
2
1
4 hj sin(2)ji ;
2
1
and D 1, so
?
2
2
2
p
3 :
2
In words, any amount of energy transferred from mode a to mode a is dissipated there almost
immediately, with no time to backscatter to mode a . Notice that, unlike the original system in
(5.1,5.2), the approximate system in (6.1,6.2) possesses a Maxwell demon which strictly forbids
transfer of energy from mode a to mode a . In view of the ordering between the 's occurring as
D 1, this leads to no practical diculty in the regime where the system in (6.1,6.2) is relevant.
However, an interesting consequence of the presence of the Maxwell demon is that the system in
(6.1,6.2) predicts h i h i in the range D 1 (see (6.4) and (6.5)). In other words, the
approximate system never thermalizes and the energy keeps owing the same way from a to a
even if a becomes the lower temperature mode.
1
2
1
2
2
2
1
2
1
1
1
20
2
8 Conclusions and Further Extensions
A system with only two free modes is, presumably, the simplest model for energy transfer. Here we
have introduced one such model, with a number of pleasant properties:
The unforced, undissipated system has the Hamiltonian structure and conserved quantities
typical of more general dispersive systems.
The forcing, in the form of white noise, permits a strict control of the energy ux through the
system.
There is a single free parameter, the quotient of the damping coecient to the amplitude of
the forcing times the square root of the strength of the nonlinearity.
Many properties of the statistically steady states of the system can be easily estimated. Yet
there is one important output {the mean amplitude of the forced mode{ which does not fall
o a preliminary analysis.
A numerical study of the system shows a clear transition between Gaussian, near equilibrium
behavior at high temperatures, to highly intermittent, non Gaussian behavior when the rate of
dissipation is high. Both behaviors can be understood even quantitatively from an analysis of the
Fokker-Planck equation of the system. We think that this \solvable" model may shed light on
similar transitions to intermittency taking place in many [far more complex] turbulent scenarios.
There are two obvious extensions that we plan to investigate in the near future: the inclusion
of an intermediate, inertial range (modes that are neither forced nor damped), and the eects of
non-resonant interactions. Both extensions are necessary if one hopes to fully understand the rich
phenomenology of wave turbulence.
Acknowledgments
The work of P. A. Milewski was partially supported by NSF grant DMS-9401405; the work of E.
G. Tabak was partially supported by NSF grant DMS-9501073; and the work of E. Vanden Eijnden
was partially supported by NSF grant DMS-9510356.
21
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22